This paper presents and analyzes an immersed finite element (IFE) method for solving Stokes interface problems with a piecewise constant viscosity coefficient that has a jump across the interface. In the method, the triangulation does not need to fit the interface and the IFE spaces are constructed from the traditional $CR$-$P_0$ element with modifications near the interface according to the interface jump conditions. We prove that the IFE basis functions are unisolvent on arbitrary interface elements and the IFE spaces have the optimal approximation capabilities, although the proof is challenging due to the coupling of the velocity and the pressure. The stability and the optimal error estimates of the proposed IFE method are also derived rigorously. The constants in the error estimates are shown to be independent of the interface location relative to the triangulation. Numerical examples are provided to verify the theoretical results.
翻译:本文介绍并分析了一种沉浸的有限要素(IFE) 方法,用一个可以跳过界面的片状常态粘度系数来解决斯托克斯界面问题。 在这种方法中,三角格不需要适应界面,而IFE空间则根据传统的 $CR$-$P_0 元元素,并根据界面跳跃条件在界面附近进行修改后建造。我们证明IFE 基函数对任意界面元素是未解的,而IFE 空格具有最佳近似能力,尽管由于速度和压力的混合,证据是具有挑战性的。还严格地计算了拟议IFE方法的稳定性和最佳误差估计值。错误估计中的常数显示与三角对界面的位置无关。提供了数字示例,以核实理论结果。